Axiom Of Choice

We have the following indirect implication of form equivalence classes:

264 $\Rightarrow$ 77 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
264 $\Rightarrow$ 202	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
202 $\Rightarrow$ 40	clear
40 $\Rightarrow$ 43	Consistency results for $ZF$, Jensen, R.B. 1967, Notices Amer. Math. Soc. On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
43 $\Rightarrow$ 77	The Axiom of Choice, Jech, 1973b, page 23

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
264:	$H(C,P)$: Every connected relation $(X,R)$ contains a $\subseteq$-maximal partially ordered set.
202:	$C(LO,\infty)$: Every linearly ordered family of non-empty sets has a choice function.
40:	$C(WO,\infty)$: Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
43:	$DC(\omega)$ (DC), Principle of Dependent Choices: If $S$ is a relation on a non-empty set $A$ and $(\forall x\in A) (\exists y\in A)(x S y)$ then there is a sequence $a(0), a(1), a(2), \ldots$ of elements of $A$ such that $(\forall n\in\omega)(a(n)\mathrel S a(n+1))$. See Tarski [1948], p 96, Levy [1964], p. 136.
77:	A linear ordering of a set $P$ is a well ordering if and only if $P$ has no infinite descending sequences. Jech [1973b], p 23.

Comment:

Back

Implication	Reference
264 \(\Rightarrow\) 202	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
202 \(\Rightarrow\) 40	clear
40 \(\Rightarrow\) 43	Consistency results for $ZF$, Jensen, R.B. 1967, Notices Amer. Math. Soc. On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
43 \(\Rightarrow\) 77	The Axiom of Choice, Jech, 1973b, page 23

Howard-Rubin Number	Statement
264:	\(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set.
202:	\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has a choice function.
40:	\(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
43:	\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136.
77:	A linear ordering of a set \(P\) is a well ordering if and only if \(P\) has no infinite descending sequences. Jech [1973b], p 23.